Fast and Exact Discrete Geodesic Computation Based on Triangle-Oriented Wavefront Propagation

Computing discrete geodesic distance over triangle meshes is one of the fundamental problems in computational geometry and computer graphics. In this problem, an effective window pruning strategy can significantly affect the actual running time. Due to its importance, we conduct an in-depth study of window pruning operations in this paper, and produce an exhaustive list of scenarios where one window can make another window partially or completely redundant. To identify a maximal number of redundant windows using such pairwise cross checking, we propose a set of procedures to synchronize local window propagation within the same triangle by simultaneously propagating a collection of windows from one triangle edge to its two opposite edges. On the basis of such synchronized window propagation, we design a new geodesic computation algorithm based on a triangle-oriented region growing scheme. Our geodesic algorithm can remove most of the redundant windows at the earliest possible stage, thus significantly reducing computational cost and memory usage at later stages. In addition, by adopting triangles instead of windows as the primitive in propagation management, our algorithm significantly cuts down the data management overhead. As a result, it runs 4-15 times faster than MMP and ICH algorithms, 2-4 times faster than FWP-MMP and FWP-CH algorithms, and also incurs the least memory usage

Fast and Memory-Efficient Voronoi Diagram Construction on Triangle Meshes

© 2017 The Author(s) Computer Graphics Forum © 2017 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. Geodesic based Voronoi diagrams play an important role in many applications of computer graphics. Constructing such Voronoi diagrams usually resorts to exact geodesics. However, exact geodesic computation always consumes lots of time and memory, which has become the bottleneck of constructing geodesic based Voronoi diagrams. In this paper, we propose the window-VTP algorithm, which can effectively reduce redundant computation and save memory. As a result, constructing Voronoi diagrams using the proposed window-VTP algorithm runs 3–8 times faster than Liu et al.'s method [LCT11] , 1.2 times faster than its FWP-MMP variant and more importantly uses 10–70 times less memory than both of them

Fast Wavefront Propagation (FWP) for Computing Exact Geodesic Distances on Meshes

Computing geodesic distances on triangle meshes is a fundamental problem in computational geometry and computer graphics. To date, two notable classes of algorithms, the Mitchell-Mount-Papadimitriou (MMP) algorithm and the Chen-Han (CH) algorithm, have been proposed. Although these algorithms can compute exact geodesic distances if numerical computation is exact, they are computationally expensive, which diminishes their usefulness for large-scale models and/or time-critical applications. In this paper, we propose the fast wavefront propagation (FWP) framework for improving the performance of both the MMP and CH algorithms. Unlike the original algorithms that propagate only a single window (a data structure locally encodes geodesic information) at each iteration, our method organizes windows with a bucket data structure so that it can process a large number of windows simultaneously without compromising wavefront quality. Thanks to its macro nature, the FWP method is less sensitive to mesh triangulation than the MMP and CH algorithms. We evaluate our FWP-based MMP and CH algorithms on a wide range of large-scale real-world models. Computational results show that our method can improve the speed by a factor of 3-10

Fast and exact geodesic computation using Edge-based Windows Grouping.

Computing discrete geodesic distance over triangle meshes is one of the fundamental problems in computational geometry and computer graphics. As the “Big Data Era” arrives, a fast and accurate solution to the geodesic computation problem on large scale models with constantly increasing resolutions is desired. However, it is still challenging to deal with the speed, memory cost and accuracy of the geodesic computation at the same time. This thesis addresses the aforementioned challenge by proposing the Edge- based Windows Grouping (EWG) technique. With the local geodesic information encoded in a “window”, EWG groups the windows based on the mesh edges and processes them together. Thus, the interrelationships among the grouped windows can be utilized to improve the performance of geodesic computation on triangle meshes. Based on EWG, a novel exact geodesic algorithm is proposed in this thesis, which is fast, accurate and memory-efficient. This algorithm computes the geodesic distances at mesh vertices by propagating the geodesic information from the source over the entire mesh. Its high performance comes from its low computational redundancy and management overhead, which are both introduced by EWG. First, the redundant windows on an edge can be removed by comparing its distance with those of the other windows on the same edge. Second, the windows grouped on an edge usually have similar geodesic distances and can be propagated in batches efficiently. To the best of my knowledge, the proposed exact geodesic algorithm is the fastest and most memory-efficient one among all existing methods. In addition, the proposed exact geodesic algorithm is revised and employed to construct the geodesic-metric-based Voronoi diagram on triangle meshes. In this application, the geodesic computation is the bottleneck in both the time and memory costs. The proposed method achieves low memory cost from the key observation that the Voronoi diagram boundaries usually only cross a minority of the meshes’ triangles and most of the windows stored on edges are redundant. As a result, the proposed method resolves the memory bottleneck of the Voronoi diagram construction without sacrificing its speed

NeuroGF: A Neural Representation for Fast Geodesic Distance and Path Queries

Geodesics are essential in many geometry processing applications. However, traditional algorithms for computing geodesic distances and paths on 3D mesh models are often inefficient and slow. This makes them impractical for scenarios that require extensive querying of arbitrary point-to-point geodesics. Although neural implicit representations have emerged as a popular way of representing 3D shape geometries, there is still no research on representing geodesics with deep implicit functions. To bridge this gap, this paper presents the first attempt to represent geodesics on 3D mesh models using neural implicit functions. Specifically, we introduce neural geodesic fields (NeuroGFs), which are learned to represent the all-pairs geodesics of a given mesh. By using NeuroGFs, we can efficiently and accurately answer queries of arbitrary point-to-point geodesic distances and paths, overcoming the limitations of traditional algorithms. Evaluations on common 3D models show that NeuroGFs exhibit exceptional performance in solving the single-source all-destination (SSAD) and point-to-point geodesics, and achieve high accuracy consistently. Moreover, NeuroGFs offer the unique advantage of encoding both 3D geometry and geodesics in a unified representation. Code is made available at https://github.com/keeganhk/NeuroGF/tree/master

Parallel and scalable heat methods for geodesic distance computation

In this paper, we propose a parallel and scalable approach for geodesic distance computation on triangle meshes. Our key observation is that the recovery of geodesic distance with the heat method from [Crane et al. 2013] can be reformulated as optimization of its gradients subject to integrability, which can be solved using an efficient first-order method that requires no linear system solving and converges quickly. Afterward, the geodesic distance is efficiently recovered by parallel integration of the optimized gradients in breadth-first order. Moreover, we employ a similar breadth-first strategy to derive a parallel Gauss-Seidel solver for the diffusion step in the heat method. To further lower the memory consumption from gradient optimization on faces, we also propose a formulation that optimizes the projected gradients on edges, which reduces the memory footprint by about 50%. Our approach is trivially parallelizable, with a low memory footprint that grows linearly with respect to the model size. This makes it particularly suitable for handling large models. Experimental results show that it can efficiently compute geodesic distance on meshes with more than 200 million vertices on a desktop PC with 128GB RAM, outperforming the original heat method and other state-of-the-art geodesic distance solvers